I have two accounts, an administrator and a simple user. I can capture the administrator's cookie and use it in a simple user request, I find myself logged in with the administrator user and then log in to the site as an administrator user. Can I consider this vulnerability a privilege escalation? How should the cookie be mitigated?
From the way you are wording this question and your comments, it sounds like this is a hypothetical. So I will treat it as such.
What you've described in your question is privilege escalation: you are starting with a user account, stealing a session cookie, and then using it to escalate your privileges to admin. However this may not be how the vulnerability is classified. In your example, there is a user and an admin account. In your comment you said
"Suppose I can capture [the admin's] cookie with privileges for example through an XSS attack."
If the site is vulnerable to XSS, then the vulnerability here would be XSS. While you are performing privilege escalation by grabbing the admin session, you are using XSS to do it. Fixing this vulnerability would be done by patching the vector for code injection.
If this does not answer what you are asking, please clarify what you are looking for. Your question is a little vague and seems to be asking about a definition.